How to match cycles texture with displacement modifier?
There are two forms of displacement texture. One is the Displacement modifier, which is a modifier and runs off of a texture. You can edit the texture by clicking the checkerboard-like icon when you have the Displacement modifier open. For that, you can choose a noise texture if you want, and there are some parameters you can apply. You could also choose the "image" option, to load an image of the noise texture you're using.

A better option for you, though, would be to use the "displacement" option in the material node editor. The final node in the editor has three inputs: Surface, Volume, and Displacement. Plug your noise texture into a Displacement node (from the Vector section), and connect that to the Displacement input. Then, in the settings for the material itself, set the Displacement mode to "Displacement Only" or "Displacement and Bump". The change won't be visible in layout mode, but it will be visible in render mode.

EDIT: To help with clarity, here's a screenshot indicating the "edit texture" option when using the Displacement modifier. To be clear, the "checkerboard icon" I was referring to is circled in red.



Here's a screenshot indicating the menu within the texture edit panel where you can control what type of texture you're using. It does not interface with the node editor, to my knowledge, but you can use an image of your noise texture. To be clear, the menu is circled in red.



Here's a screenshot presenting the node setup I suggested. There's no red circle in this one; I hope you're able to figure it out anyway.



For the "Displacement Only" and "Displacement and Bump" settings, go to the Material edit pane (in the panel on the right, where you found the modifiers), then the "Settings" section, and then the "Surface" section within that section. Please note that this option will not appear if you do not have "Cycles" selected as your renderer.